Find The Sum of the First n Bouncy Numbers

Jelly, 10 8 bytes

ṢeṚƬ¬µ#S

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How it works

ṢeṚƬ¬µ#S  Main link. No arguments.

      #   Read an integer n from STDIN and call the chain to the left with argument
          k = 0, 1, 2, ... until n of them return a truthy result.
          Yield the array of successful values of k.
     µ    Monadic chain. Argument: k (integer)
Ṣ           Sort, after promoting k to its digit array.
  ṚƬ        Reverse 'til the results are no longer unique and yield unique results.
            Calling Ṛ on k promotes it to its digit array. If k = 14235, the 
            result is [14235, [5,3,2,4,1], [1,4,2,3,5]].
 e          Check if the result to the left appears in the result to the right.
    ¬       Negate the resulting Boolean.
       S  Take the sum.

Pyth, 10 bytes

s.f!SI#_B`

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How it works?

s.f!SI#_B` – Full program. Takes an integer Q from STDIN and outputs to STDOUT.
 .f        – Find the first Q positive integers that satisfy a certain condition.
   !SI#_B  – The condition. Returns true for bouncy numbers only.
       _B` – Cast the number to a string and bifurcate (pair) it with its reverse.
      #    – Filter-keep those...
     I     – That are invariant under...
    S      – Sorting.
           – To clarify, I (invariant) is a Pyth operator that takes two inputs, a 
             function and a value and checks whether function(value) == value, so
             this is technically not a built-in.
   !       – Logical not. The empty list gets mapped to true, other values to false.
s          – Sum.

K (ngn/k), 37 bytes

{+/x{{(a~&\a)|a~|\a:10\x}(1+)/x+1}\0}

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{ } is a function with argument x

x{ }\0 applies the {} on 0 x times, preserving the intermediate results

(1+) is the successor function

{ }(1+)/x+1 applies the successor function starting from x+1 until the {} returns true

10\x are the decimal digits of x

a: assign to a

|\ is the max-scan (partial maxima) of a

&\ analogously, is the min-scan

a~|\a does a match its max-scan?

| or

a~&\a its min-scan?

+/ sum